The potential use of the wavelet-decomposition method developed by the authors for the analysis of geomagnetic data and cosmic ray variations is studied. With the use of adaptive threshold functions, the method allows the isolation of nonstationary, short-period (from 1 × 10–3 Hz and greater) variations in the data and the estimation of their parameters. Data from a network of ground-based magnetometers (www.inrtermagnet.org) and neutron monitors (http://cosray.unibe.ch/) were used in the work. The advantages of the method are shown on the example of magnetic storms on July 9 and September 27, 2017; the effectiveness of the method for the detection of low-amplitude anomalous variations in heterogeneous data has been proven experimentally. The dynamics of variations in the geomagnetic field at meridionally located stations and in the auroral zone is considered in detail; patterns of the occurrence and propagation of geomagnetic disturbances preceding and accompanying the intervals of magnetic storms are obtained. The application of the method made it possible to detect clearly and estimate weak short-term increases in geomagnetic activity observed against the background of increased cosmic ray intensity preceding the onset of magnetic storms. It is noted that the identified geomagnetic disturbances occurred synchronously at stations from high latitudes to the equator and correlated with the periods of southward turns of the IMF Bz component and increased auroral activity. During these intervals, cosmic rays exhibited low-amplitude Forbush effects, which were detected with this method.